# Is pocket algorithm optimal?

## Abstract

The pocket algorithm is considered able to provide for any classification problem the weight vector which satisfies the maximum number of input-output relations contained in the training set. A proper convergence theorem ensures the achievement of an optimal configuration with probability one when the number of iterations grows indefinitely. In the present paper a new formulation of this theorem is given; a rigorous proof corrects some formal and substantial errors which invalidate previous theoretical results. In particular it is shown that the optimality of the asymptotical solution is ensured only if the number of permanences for the pocket vector lies in a proper interval of the real axis which bounds depend on the number of iterations.

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## References

- 1.Cybenko, G. Approximation by superpositions of a sigmoidal function.
*Mathematics of Control, Signals, and Systems***2**(1989), 303–314.Google Scholar - 2.Hornik, K., Stinchcombe, M., and White, H. Multilayer feedforward networks are universal approximators.
*Neural Networks***2**(1989), 359–366.Google Scholar - 3.Blum, A., AND Rivest, R. L. Training a 3-node neural network is NP-complete. In
*Proceedings of the 1988 Workshop on Computational Learning Theory*(Cambridge, MA, 1988), D. Haussler and L. Pitt, Eds., Morgan Kaufmann, pp. 9–18.Google Scholar - 4.Hertz, J., Krogh, A., AND Palmer, R. G.
*Introduction to the Theory of Neural Computation*. Redwood City, CA: Addison-Wesley, 1991.Google Scholar - 5.Gallant, S. I.
*Neural Networks Learning and Expert Systems*. Cambridge, MA: MIT Press, 1993.Google Scholar - 6.
- 7.Minsky, M., AND Papert, S.
*Perceptrons: An Introduction to Computational Geometry*. Cambridge, MA: MIT Press, 1969.Google Scholar - 8.Ho, Y.-C., AND Kashyap, R. L. An algorithm for linear inequalities and its applications.
*IEEE Transactions on Electronic Computers***14**(1965), 683–688.Google Scholar - 9.Khachiyan, L. G. A polynomial algorithm in linear programming.
*Soviet Mathematics Doklady***20**(1979), 191–194.Google Scholar - 10.Mansfield, A. J. Comparison of perceptron training by linear programming and by the perceptron convergence procedure. In
*Proceedings of the International Joint Conference on Neural Networks*(Seattle, WA, 1991), pp. II-25–II-30.Google Scholar - 11.Gallant, S. I. Perceptron-based learning algorithms.
*IEEE Transactions on Neural Networks***1**(1990), 179–191.Google Scholar - 12.Mézard, M., AND Nadal, J.-P. Learning in feedforward layered networks: The tiling algorithm.
*Journal of Physics A***22**(1989), 2191–2203.Google Scholar - 13.Frean, M. The upstart algorithm: A method for constructing and training feed-forward neural networks.
*Neural Computation***2**(1990), 198–209.Google Scholar - 14.Muselli, M. On sequential construction of binary neural networks. To appear on
*IEEE Transactions on Neural Networks*.Google Scholar - 15.